From 1d6a9cf1441bd6d478977715d82031e77c20ce5c Mon Sep 17 00:00:00 2001 From: Marshall Lochbaum Date: Mon, 26 Oct 2020 15:46:38 -0400 Subject: Don't include &run in documentation REPL links: it's no longer used --- docs/doc/train.html | 22 +++++++++++----------- 1 file changed, 11 insertions(+), 11 deletions(-) (limited to 'docs/doc/train.html') diff --git a/docs/doc/train.html b/docs/doc/train.html index c7f9eeea..d4eff715 100644 --- a/docs/doc/train.html +++ b/docs/doc/train.html @@ -9,15 +9,15 @@

BQN's trains are the same as those of Dyalog APL, except that Dyalog is missing the minor convenience of BQN's Nothing (·). There are many Dyalog-based documents and videos trains you can view on the APL Wiki.

2-train, 3-train

Trains are an adaptation of the mathematical convention that, for example, two functions F and G can be added to get a new function F+G that applies as (F+G)(x) = F(x)+G(x). With a little change to the syntax, we can do exactly this in BQN:

-↗️
    (⊢+⌽) ↕5
+↗️
    (⊢+⌽) ↕5
 ⟨ 4 4 4 4 4 ⟩
 

 
So given a list of the first few natural numbers, that same list plus its reverse gives a list of just one number repeated many times. I'm sure if I were Gauss I'd be able to find some clever use for that fact. The mathematical convention extends to any central operator and any number of function arguments, which in BQN means we use any three functions, and call the train with a left argument as well—the only numbers of arguments BQN syntax allows are 1 and 2.

-↗️
    7 (+≍-) 2
+↗️
    7 (+≍-) 2
 ⟨ 9 5 ⟩
 

 
Here Couple (≍) is used to combine two units into a list, so we get seven plus and minus two. It's also possible to leave out the leftmost function of a train, or replace it with ·. In this case the function on the right is called, then the other function is called on its result—it's identical to the mathematical composition ∘, which is also part of BQN.

-↗️
    (∾⌽) "ab"‿"cde"‿"f"
+↗️
    (∾⌽) "ab"‿"cde"‿"f"
 "fcdeab"
     (·∾⌽) "ab"‿"cde"‿"f"
 "fcdeab"
@@ -48,24 +48,24 @@
 

 
So—although not all trains simplify so much—this confusing train is just {𝕩>¯1»⌈`𝕩}! Why would I write it in such an obtuse way? To someone used to working with trains, the function (⊢>¯1»⌈`) isn't any more complicated to read: ⊢ in an argument position of a train just means 𝕩 while ⌈` will be applied to the arguments. Using the train just means slightly shorter code and two fewer 𝕩s to trip over.

 
This function's argument is the self-classify ⊐ of some list (in fact this technique also works on the self-indices 𝕩⊐𝕩). Self-classify moves along its argument, giving each major cell a number: the first unused natural number if that value hasn't been seen yet, and otherwise the number chosen when it was first seen. It can be implemented as ∊⊐⊢, another train!

-↗️
    ⊢ sc ← ⊐ "tacittrains"
+↗️
    ⊢ sc ← ⊐ "tacittrains"
 ⟨ 0 1 2 3 0 0 4 1 3 5 6 ⟩
 

 
Each 't' is 0, each 'a' is 1, and so on. We'd like to discard some of the information in the self-classify, to just find whether each major cell had a new value. Here are the input and desired result:

-↗️
    sc ≍ ∊ "tacittrains"
+↗️
    sc ≍ ∊ "tacittrains"
 ┌─                       
 ╵ 0 1 2 3 0 0 4 1 3 5 6  
   1 1 1 1 0 0 1 0 0 1 1  
                         ┘
 

 
The result should be 1 when a new number appears, higher than all the previous numbers. To do this, we first find the highest previous number by taking the maximum-scan ⌈` of the argument, then shifting to move the previous maximum to the current position. The first cell is always new, so we shift in a ¯1, so it will be less than any element of the argument.

-↗️
    ¯1 » ⌈`sc
+↗️
    ¯1 » ⌈`sc
 ⟨ ¯1 0 1 2 3 3 3 4 4 4 5 ⟩
     (¯1»⌈`) sc
 ⟨ ¯1 0 1 2 3 3 3 4 4 4 5 ⟩
 

 
Now we compare the original list with the list of previous-maximums.

-↗️
    sc > ¯1»⌈`sc
+↗️
    sc > ¯1»⌈`sc
 ⟨ 1 1 1 1 0 0 1 0 0 1 1 ⟩
     (⊢>¯1»⌈`) sc
 ⟨ 1 1 1 1 0 0 1 0 0 1 1 ⟩
@@ -73,24 +73,24 @@
 
Composing trains

 
The example above uses a train with five functions: an odd number. Trains with an odd length are always composed of length-3 trains, and they themselves are composed the same way as subject expressions: an odd-length train can be placed in the last position of another train without parentheses, but it needs parentheses to go in any other position.

 
But we also saw the length-2 train ∾⌽ above. Even-length trains consist of a single function (∾) applied to a function or odd-length train (⌽); another perspective is that an even-length train is an odd-length train where the left argument of the final (leftmost) function is left out, so it's called with only a right argument. An even-length train always needs parentheses if it's used as one of the functions in another train. However, it can also be turned into an odd-length train by placing · at the left, making the implicit missing argument explicit. After this it can be used at the end of an odd-length train without parentheses. To get some intuition for even-length trains, let's look at an example of three functions used together: the unique (⍷) sorted (∧) absolute values (|) of an argument list.

-↗️
    ⍷∧| 3‿4‿¯3‿¯2‿0
+↗️
    ⍷∧| 3‿4‿¯3‿¯2‿0
 ⟨ 0 2 3 4 ⟩
 

 
If it doesn't have to be a function, it's easiest to write it all out! Let's assume we want a tacit function instead. With three one-argument functions, we can't use a 3-train, as the middle function in a 3-train always has two arguments. Instead, we will compose the functions with 2-trains. Composition is associative, meaning that this can be done starting at either the left or the right.

-↗️
    ((⍷∧)|) 3‿4‿¯3‿¯2‿0
+↗️
    ((⍷∧)|) 3‿4‿¯3‿¯2‿0
 ⟨ 0 2 3 4 ⟩
     (⍷(∧|)) 3‿4‿¯3‿¯2‿0
 ⟨ 0 2 3 4 ⟩
 

 
We might make the first train above easier to read by using Atop (∘) instead of a 2-train. Atop is a 2-modifier, so it doesn't need parentheses when used in a train. The second train can also be changed to ⍷∧∘| in the same way, but there is another option: the rightmost train ∧| can be expanded to ·∧|. After this it's an odd-length train in the last position, and doesn't need parentheses anymore.

-↗️
    (⍷∘∧|) 3‿4‿¯3‿¯2‿0
+↗️
    (⍷∘∧|) 3‿4‿¯3‿¯2‿0
 ⟨ 0 2 3 4 ⟩
     (⍷·∧|) 3‿4‿¯3‿¯2‿0
 ⟨ 0 2 3 4 ⟩
 

 
These two forms have a different emphasis, because the first breaks into subfunctions ⍷∘∧ and | and the second into ⍷ and ∧|. It's more common to use ⍷∘∧ as a unit than ∧|, so in this case ⍷∘∧| is probably the better train.

 
Many one-argument functions strung together is a major weakness for train syntax. If there are many such functions it's probably best to stick with a block function instead!

-↗️
    {⍷∧|𝕩} 3‿4‿¯3‿¯2‿0
+↗️
    {⍷∧|𝕩} 3‿4‿¯3‿¯2‿0
 ⟨ 0 2 3 4 ⟩
 

 
In our example, there aren't enough of these functions to really be cumbersome. If ⍷∘∧ is a common combination in a particular program, then the train ⍷∘∧| will be more visually consistent and make it easier to use a utility function for ⍷∘∧ if that's wanted in the future.

-- 
cgit v1.2.3